Introduction to radiative
transfer
Robert Hudson
Electromagnetic spectrum
Schematic of a wave
WAVELENGTH
• DISTANCE BETWEEN SUCCESSIVE PEAKS
• GIVEN THE SYMBOL λ
• MANY UNITS USED :
•
(A) MICRON, 10-6 METERS
•
(B) NANOMETER, 10-9 METERS
•
(C) ANGSTROM, 10-10 METERS
FREQUENCY
Defined as the number of ma xima that pass an observer
per second
Given the symbol ~
c  ~
The term wave number is also used. It is the reciprocal
of the wavelengt h. It is usually given the symbol  .
Caution - the use of  and ~ are reversed in many
textbooks.
VELOCITY
• WAVE VELOCITY IS DEFINED AS THE DISTANCE A
PEAK MOVES IN ONE SECOND.
• IN VACUO THE WAVE VELOCITY OF AN
ELECTROMAGNETIC WAVE IS 2.997925x108 METERS
PER SECOND
• FOR THOSE OF YOU WHO PREFER ENGLISH UNITS,
THE VELOCITY OF LIGHT IS 1.80262x1012
FURLONGS PER FORTNIGHT.
• THE WAVE VELOCITY IS GIVEN THE SYMBOL c.
Planck’s spectral distribution law
• Planck introduced in 1901 his hypothesis
of quantized oscillators in a radiating body.
• He derived an expression for the
hemispherical blackbody spectral radiative
flux
BB
F
2
r
2
m

c
2h
exp( h /kB T) 1
3
Where h is Planck’s constant, mr is the real
index of refraction, kB is Boltzmann’s constant
Comparison of solar and earth’s
blackbody intensity
UV-VISIBLE SOLAR SPECTRUM
INFRARED FLUX FROM EARTH
SAHARA
Absorption in molecular lines and
bands
• Molecules have three types of energy levels electronic, vibrational, and rotational
• Transitions between electronic levels occur
mainly in the ultraviolet
• Transitions between vibrational levels visible/near IR
• Transitions between rotational levels - far IR/
mm wave region
• O2 and N2 have essentially no absorption in
the IR
• 4 most important IR absorbers H2O, CO2, O3,
CH4
Vibrational levels
• Consider a diatomic molecule. The two
atoms are bound together by a force, and
can oscillate along the axis of the
molecule.
• The force between the two atoms is given
2
by
dx
F  k.x  m
dt
2
• The solution of which is
x  x0 sin( 2 0t  )
Vibrational levels
• 0 is known as the vibrational frequency
1
0 
2
k
m
• theoretically 0 can assume all values
• However in quantum mechanics these
values
must be discrete
E  h 0 (v 1/2)
• v is the vibrational quantum number
Vibrational levels
• In general k depends on the separation of
the atoms and we have an ‘anharmonic
oscillator’
E  hc  e (v  1 / 2)  hc  e xe (v  1 / 2)  hc  e ye (v  1 / 2)
2
3
Schematic of vibrational levels
Various forms of molecular vibration

Rotational levels
Consider a diatomic molecule with different
atoms of mass m1 and m2, whose distance from
the center of mass are r1 and r2 respectively
•

o
m1
r1
| r2 
o
m2
• The moment of inertia of the system about the
center of mass is:
Imr m r
2
11
2
2 2
Rotational
levels
The classical expression for energy of
rotation is
•
I 2 L2
Er 

where L angular momentum
2
2I
2
 h 
L    J ( J  1)
 2 
2
• where J is the rotational quantum number
2
1  h 
E(J ) 

 J ( J  1)  h c B J ( J  1)
2 I  2 
h
B 
the rotational constant
2
8 c I
Vibrating Rotator
• If there were no interaction between the
rotation and vibration, then the total energy of a
quantum state would be the sum of the two
energies. But there is, and we get
E (v, J )
2
  0 (v  1 / 2)   0 xe (v  1 / 2)
hc
2
2
 Bv J ( J  1)  Dv J ( J  1)
• The wavenumber of a spectral line is given
by the difference of the term values of the two
states
Energy levels of a vibrating rotator
Fine structure in HCl
Intensity distribution in bands of HCl
Line broadening
• Classical theory leads to the following
equation for the shape of a line (transition). It
is called the Lorentz profile
 / 4
 L ( ) 
 ( 0  ) 2  ( / 4 ) 2 
Since the Lorentz profile is normalized we
find by integrating over all frequencies

 d
0
2
res
n
e
( ) 
4me 0c
Line broadening
• The line width (full width at half maximum) of the
Lorentz profile is the damping parameter, .
• For an isolated molecule the damping parameter
can be interpreted as the inverse of the lifetime of
the excited quantum state.
• This is consistent with the Heisenberg Uncertainty
Principle
h
Et 
2
h
1
t 

2 .h 2
• If absorption line is dampened solely by the natural
lifetime of the state this is natural broadening

Pressure broadening
• For an isolated molecule the typical natural
lifetime is about 10-8 s, 5x10-4 cm-1 line width
• Collisions between molecules can shorten this
lifetime
• These collisions can be viewed as ‘billiard ball’
reactions, or as the overlapping of the potential
fields of the two molecules.
• The collision process leads to a Lorentz line
shape.
Pressure broadening
• Clearly the line width will depend on the
number of collisions per second,i.e. on the
number density of the molecules
(Pressure) and the relative speed of the
molecules (the square root of the
temperature)
nv rel
n T
 L  L ( STP)
  L ( STP)
nL v rel ( STP)
nL T0
Doppler broadening
• Second major source of line broadening
• Molecules are in motion when they absorb. This
causes a change in the frequency of the
incoming radiation as seen in the molecules
frame of reference (Doppler effect)
• Let the velocity be v, and the incoming
frequency be , then
  
'
v cos 

v v cos 
v
 
  (1  cos  )
c
c
Doppler broadening
• In the atmosphere the molecules are moving with
velocities determined by the Maxwell Boltzmann
distribution
1/ 2
 m 

f ( v X )dv X  
 2k BT 
where v 0  2k BT / m
exp(  v 2X / v 02 )dv X
Doppler broadening
•The cross section at a frequency  is the sum of all
line of sight components

 n ( )   dv x f ( v x ) n  (1  v x / c)

1/ 2  
 m 

 
 2k BT 
2
2
v
/
v

exp(
dv
0 ) n (  v x / c )
x
 x

1/ 2
 m 
2 2
2
2


exp  c (  0 ) / v 0 v0
S

 2k BT 


Doppler broadening
•
We now define the Doppler width as
 D  v 0 v0 / c
 n ( )  S  D ( ) 
S
D

exp  (  0 ) 2 /  D2

Comparison of line shapes
The extinction law
Extinction Law
• The extinction law can be written as
dI   k ( ) I ds
• The constant of proportionality is defined as the
extinction coefficient. k can be defined by the length of
the absorbing path with the gas at one atmosphere
pressure
dI
k ( ) 
I ds
(m 1 )
Optical depth
• Normally we are interested in the total
extinction over a finite distance (path length)
s
s
s
0
0
0
 s ( )   ds' k ( )   ds' km ( )    ds' kn ( )n
Where S() is the extinction optical depth
• The integrated form of the extinction equation
becomes
I ( s,  )  I (0,  ) exp   s ( )
Extinction = scattering + absorption
• Extinction really consists of two distinct
processes, scattering and absorption, hence
 s ( )  sc ( )  a ( )
where
s
 sc ( )    ds  ( , s ' )
i
i
i
0
s
 a ( )    ds  ( , s' )
i
i
0
i
Differential equation of radiative
transfer
• We must now add the process called emission.
• We introduce an emission coefficient, jν
• Combining the extinction law with the definition
of the emission coefficient
dI   k ( ) I ds  j ds
noting that
k ( )ds  d s
dI
j
  I 
d s
k ( )
Differential equation of radiative
transfer
• The ratio j/k() is known as the source
function,
j
S 
k ( )
dI
  I  S
d s
This is the differential equation of radiative
transfer
Scattering
• Two types of scattering are considered –
molecular scattering (Rayleigh) and
scattering from aerosols (Mie)
• The equation for Rayleigh scattering can be
written as
4

•
RAY
n
8 2  2
( ) 
   p
3   
Where α is the polarizability

Rayleigh scattering
, nm
, cm2
, surface
Exp(-)
300
6.00 E-26
1.2
0.301
400
1.90 E-26
0.38
0.684
600
3.80 E-27
0.075
0.928
1000
4.90 E-28
0.0097
0.990
10,000
4.85 E-32
9.70 E-7
0.999
• Sky appears blue at noon, red at sunrise and
sunset - why?
Phase diagram for Rayleigh scattering
Mie-Debye scattering
• For particles which are not small compared with
the wavelength one has to deal with multiple
waves from different molecules/atoms within the
particle
• Forward moving waves tend to be in phase and
this gives a large resultant amplitude.
• Backward waves tend to be out of phase and
this results in a small resultant amplitude
• Hence the scattering phase function for a
particle has a much larger forward component
(forward peak) than the backward component
Phase diagrams for aerosols
Differential Equation of Radiative Transfer
• Introduce two additional parameters. B, the
Planck function, and a , the single scattering
albedo (the ratio of the scattering cross section
to the extinction coefficient).
• The complete time-independent radiative
transfer equation which includes both
scattering and absorption is
dI
a( )
ˆ ', 
ˆ )I
 I 1  a( )B (T )
d

'
p
(



d s
4 4
Solution for Zero Scattering
• If there is no scattering, e.g. in the thermal
infrared, then the equation becomes
dI
 I  B (T)
d S

Transmittance
• For monochromatic radiation the transmittance, T, is
given simply by
T ( ;  )  e
 / 
• But now we must consider how to deal with
radiation that is not monochromatic. In this case the
integration must be made over all frequencies.
• Absorption cross section at high spectral resolution
are available in tabular form – HITRAN.
• But usually an average value over a frequency
interval is used.
Schematic for A
Transmission in spectrally complex media
• transmittance
T
1
 1     

  k u 
  e d 


 

Statistical Band Model (Goody)
• Goody studied the water-vapor bands and noted the
apparent random line positions and band strengths.
• Let us assume that the interval contains n lines
of mean separation d, i.e. =nd
• Let the probability that line i has a line strength Si
be P(Si) where

 P(S )dS  1
i
0
• P is normally assumed to have a Poisson
distribution
Statistical Band Model (Goody)
assuming a Lorentz line shape, one gets
1 / 2
 Su 
Su  
1 
 
T  exp 
 d  L  
where S is the mean line strength.
Statistical Band Model (Goody)
• We have reduced the parameters needed to calculate T
to two
S
d
and
L
d
• These two parameters are either derived by
fitting the values of T obtained from a line-by-line
calculation, or from experimental data.
weak line
 Su 

T  exp  
 d 
strong line



S
u
L

T  exp  


d


K distribution technique
K distribution technique
Two-stream Approximation- Isotropic
Scattering
• Although anisotropic scattering is more
realistic, let’s look at isotropic scattering i.e.
p=1
• The radiative transfer equations are
dI ( ,  )
a


 I ( ,  )   d 'I  ( ,  ' )
d
20

1
1
a

  d 'I ( ,  ' )  (1  a) B
20
Two-stream Approximation- Isotropic
Scattering
1

dI ( ,  )
a



 I ( ,  )   d 'I ( ,  ' )
d
20
1
a

  d 'I ( ,  ' )  (1  a ) B
20
• In the two-stream approximation we
replace the angular dependent quantities I by
their averages over each hemisphere. This
leads to the following pair of coupled
differential equations
Two-stream Approximation- Isotropic
Scattering

dI
( )
a 
a 



 I ( )  I ( )  I ( )  (1  a) B
d
2
2

dI
( )
a 
a 



 I ( )  I ( )  I ( )  (1  a) B
d
2
2
If the medium is homogeneous then a is
constant. One can now obtain analytic
solutions to these equations.  in the above
equations is the cosine of the average polar
angle. It generally differs in the two
hemispheres
Discrete Ordinate Method – Isotropic Scattering
• The solution of the isotropic scattering problem
involves the following integral over angle
1
1
1
0
 duI ( , u)   dI

1
( , u )   dI ( , u )

0
• In the two stream method we replaced the
integration over  with the simple formula
1
 duI  I
1

( )  I ( )

Discrete Ordinate Method – Isotropic Scattering
• This is obviously a crude approximation. We can improve
the accuracy by including more points in a numerical
quadrature formula
1
m
1
j 1
'
duI
(

,
u
)

w
 j I ( , u j )

• Where w’j is a quadrature weight, and uj is the discrete
ordinate
• Most commonly used radiative transfer computer codes is
DISORT – DIScreteOrdinateRadiativeTransfer